Register Allocation and Binding for Low Power. Jui-Ming Chang and Massoud Pedram. function of signal statistics and correlations, and hence is

Transcription

1 Rgistr Allotion n ining or Low Powr Jui-Ming hng n Mssou Prm Dprtmnt o Eltril Enginring-Systms Univrsity o Southrn liorni Los Angls, A 989 Astrt This ppr sris thniqu or lulting th swithing tivity o st o rgistrs shr y irnt t vlus s on th ssumption tht th joint p (proility nsity untion) o th primry input rnom vrils is known or tht suintly lrg numr o input vtors hs n givn, th rgistr ssignmnt prolm or minimum powr onsumption is ormult s minimum ost liqu ovring o n ppropritly n omptiility grph (whih is shown to trnsitivly orintl) Th prolm is thn solv optimlly (in polynomil tim) using mx-ost ow lgorithm Exprimntl rsults on- rm th viility n usulnss o th pproh in minimizing powr onsumption uring th rgistr ssignmnt phs o th hviorl synthsis pross Introution On riving tor hin th push or low powr sign is th growing lss o prsonl omputing vis s wll s wirlss ommunitions n imging systms tht mn high-sp omputtions n omplx untionlitis with low powr onsumption Anothr riving tor is tht xssiv powr onsumption is oming th limiting tor in intgrting mor trnsistors on singl hip or on multipl-hip moul Unlss powr onsumption is rmtilly ru, th rsulting ht will limit th sil pking n prormn o VLSI iruits n systms Th hviorl synthsis pross onsists o thr phss: llotion, ssignmnt n shuling Ths prosss trmin how mny instns o h rsour r n (llotion), on wht rsour omputtionl oprtion will prorm (ssignmnt) n whn it will xut (shuling) Tritionlly, hviorl synthsis ims to minimiz th numr o rsours rquir to prorm tsk in givn tim or to minimiz th xution tim or givn st o rsours It hs om nssry to vlop hviorl synthsis thniqus tht lso ount or powr issiption in th iruit This rsrh ws support in prt y th NSF's Young Invstigtor Awr unr ontrt No MIP n grnt rom th Intl orp 32n AM/IEEE Dsign Automtion onrn Prmission to opy without ll or prt o this mtril is grnt, provi tht th opis r not m or istriut or irt ommril vntg, th AM opyright noti n th titl o th pulition n its t ppr, n noti is givn tht opying is y prmission o th Assoition or omputing Mhinry To opy othrwis, or to rpulish, rquirs n/or spii prmission 995 AM /95/6 $35 This xtns th two-imnsionl optimiztion prolm to thir imnsion Th thr phss o th hviorl synthsis pross must thus moi to prou low powr iruits Unortuntly, powr issiption is strong untion o signl sttistis n orrltions, n hn is non-trministi Automti thniqus tht minimiz th swithing tivity on glolly shr usss n rgistr ls, tht slt low powr mros tht stisy th timing onstrints, tht shul oprtions to minimiz th swithing tivity rom on yl stp to nxt, t must vlop This ppr onsirs rgistr ssignmnt or low powr Most o th high-lvl synthsis systms prorm shuling o th ontrol n t ow grph (DFG) or llotion o th rgistrs n mouls n synthsis o th intronnt [][8][7] s this pproh provis timing inormtion or th llotion n ssignmnt tsks Othr systms prorm th rsour llotion n ining or shuling, in orr to provi mor pris timing inormtion vill uring th shuling [9] Eithr pproh hs its own vntgs n shortomings Th prsnt work ssums tht th shuling o th DFG hs n on n prorms th rgistr llotion or th llotion o mouls n intronntion During th rgistr llotion n ssignmnt, t vlus (rs in th t ow grph) n shr th sm physil rgistr i thir li tims o not ovrlp In th pst, rsrhrs hv propos vrious thniqus to ru th totl numr o th rgistrs us Th xisting pprohs inlu rul-s [6], gry or itrtiv [], rnh n oun [3], linr progrmming [], n grph thorti, s in th Ft systm [8], th HAL systm [6] n th EASY systm [7] Powr onsumption o wll sign rgistr sts pns minly on th totl swithing tivity o th rgistrs In mny pplitions, th t strms whih r input to th iruit hv rtin proility istriutions Vrious wys o shring rgistrs mong irnt t vlus thus prou irnt swithing tivitis in ths rgistrs This work prsnts novl wy olulting this swithing tivity s on th ssumption tht th joint p (proility nsity untion) o primry input rnom vrils is known or suintly lrg numr o input vtors hs n givn In th lttr s, th joint p n otin y sttistil mthos Atr otining th joint p o primry input vrils, th p o ny intrnl r (t vlu) in th t ow grph n th joint p o ny pir o rs (t vlus) in th t ow grph r lult y mtho tht will sri in til in th ollow-

2 ing stions Th swithing tivity on pir o rs is thn ormult in trms o th joint p o ths rs, or ltrntivly, in trms o untion o th joint p o ll primry input vrils Th li tim o h r (t vlu) in shul t ow grph is th tim uring whih th t vlu is tiv (vli) n is n y n intrvl [irth tim; th tim] A omptility grph G(V,A) or ths rs (t vlus) is thn onstrut, whr vrtis orrspon to t vlus, n thr is irt r (u,v) twn two vrtis i n only i thir orrsponing li tims o not ovrlp n th u oms or v W will show tht th unorint omptility grph or th rs (t vlus) in shul t ow grph without yls n rnhs is omprility grph (or trnsitivly orintl grph) whih isprt grph [5] This is vry usul proprty, s mny grph prolms (g mximum liqu; mximum wight k-liqu ovring, t) n solv in polynomil tim or prt grphs whil thy r NP-omplt or gnrl grphs Hving lult th swithing tivity twn pirs o rs tht oul potntilly shr th sm rgistr n givn th numr o rgistrs tht r to us, th rgistr ssignmnt prolm or minimum powr onsumption is ormult s minimum ost liqu ovring o th omptiility grph Th prolm is thn solv optimlly (in polynomil tim) using mx-ost ow lgorithm Th two prolms, lultion o th ross-r swithing tivitis (whih must prorm O(j E j) tims, whr j E j is th numr o gs in th omptiility grph) n powr minimiztion uring rgistr ssignmnt, r inpnnt Th lultion o th ross-r swithing tivitis n prorm y ny mns W prsnt on suh thniqu ltr Othr thniqus my howvr us Th powr optimiztion is prorm on th rossr swithing tivitis r known Th rminr o this ppr is orgniz s ollows: Stion 2 shows th mtho to lult th swithing tivity twn pirs o t vlus (rs) Stion 3 shows th mtho to optimiz th powr onsumption o rgistrs in th rgistr llotion phs in hviorl synthsis Stion 4 r som xmpls to monstrt th mthoology 2 Swithing Ativity lultion 2 lultion o vrious ps In mny instns, th input t strms r somwht known, n n thus sri y som proilisti istriutions (Our propos mtho pplis not only to th wll known proility istriutions, suh s joint Gussin istriution, ut lso to ritrry proility istriutions) Givn suint numr o input vtors, it is possil to n th symoli xprssions or th p's n th joint p o ll inputs using mthos in sttistis For xmpl, on wy to o this is to lult th rquny o th ourn or h vtor mong th st o input vtors, n thn prorm th intrpoltion on th sts o isrt points to otin th symoli xprssion o th joint p Altrntivly, on n work irtly with th input vtors without hving to n th symoli xprssion o th joint p, tht is, or suintly lrg numr o th input vtors, th rquny o ourn or h input vtor n srv s th vlu o th joint p or tht pttrn I w r givn th joint p o th input rnom vrils o t ow grph, thn th joint p o ny pir o vlus (rs in th t ow grph) n lult [5] W wnt to n th joint p o ny two rs Suppos tht th two rs r y = u (x ;x 2;:::;x n) n y 2 = u 2(x ;x 2;:::;x n) W n nothr (n 2) r untions y 3;y 4;:::;y n n orm systm o n qutions in n input vrils Lt's not th joint p o th n input vrils s (x ;x 2;:::;x n) I th invrs solution x = w (y ;y 2;:::;y n);x 2 = w 2(y ;y 2;:::;y n);:::;x n = w n(y ;y 2;:::;y n) n otin symolilly, thn th joint p o y ;y 2;:::;y n whih is not y (y ;y 2;:::;y n) is: whr J J (y ;y 2;:::;y n)= jj j [w (y ;y 2;:::;y n); w 2(y ;y 2;:::;y n);:::; w n(y ;y 2;:::;y n)] is th nxn invrs Join: (y ;y 2;:::;y n)= x x y x 2 x 2 y xn y yn y 2 x y 2 x 2 yn xn y 2 xn yn On w hv th (y ;y 2;:::;y n),w n lult th pirwis p o y n y 2, y y 2 (y ;y 2), s Z Z Z y y 2 (y ;y 2)= (y ;y 2;:::;y n) y 3y 4 :::y n: Th intgrtion n prorm ithr symolilly or numrilly Th numril intgrtion ovr (n 2) vrils involvs muh mor omputtion, ut is n ltrntiv pproh whih is lwys possil whnvr th symoli intgrtion ovr th (n 2) vrils is not possil In ition to th lultion o pirwis joint ps, th p o ny intrnl r is n to lult th totl swithing tivity o th st o rgistrs Suppos untion y = w(x ;x 2;:::;x n) is som r (t vlu) in th t ow grph pning on n input rnom vrils x ;x 2;:::;x n Th (umult istriution untion) o th nw rnom vril y is n s G(y) = pro(y y), whih is qul to pro(w(x ;x2;:::;x n) y) Th ov proility n vlut s: G(y) = ZZ Z A (x ;x 2;:::;x n) whr (x ;x 2;:::;x n) is th joint p o th n input rnom vrils x ;x 2;:::;x n, n A = (x ;x 2;:::;x n) j w(x ;x2;:::;x n) yg Th p o y s g(y) is thn otin y g(y) = G(y) y

3 i j k ontrol ontrol x y Mux R DMux out,mux in,r in,dmux out, R Figur : Our rgistr shring mol 22 Th powr onsumption mol Swith pitn rrs to th prout o th lo pitn n th swithing tivity o th rivr Th powr onsumption o rgistr is proportionl to th swith pitn on its input n output (s Fig ) Suppos rgistr R n shr twn thr t vlus i; j n k: W ssum tht n input multiplxor piks th vlu tht is writtn into R whil n output multiplxor ispths th stor vlu to its propr stintion Now, P (R ) / swithing(x) ( out;mux in;r )swithing(y) ( out;r in;dmux) Sin swithing(x) =swithing(y), P (R )=swithing(y) totl Not tht totl is x or givn lirry In ny s, minimizing th swithing tivity t th output o th rgistrs will minimiz th powr onsumption rgrlss o th spi lo sn t th output o th rgistrs Hr w ignor th powr onsumption intrnl to rgistrs n only onsir th xtrnl powr onsumption In th rgistr llotion phs, i svrl omptil rs r ssign to th sm rgistr R, th swithing on R will our whnvr on stor t vlu is rpl y nothr t vlu For xmpl, suppos,y,z n W r our omptil t vlus tht shr rgistr R n th rs (; Y ); (Y;Z);(Z;W) 2 A Suppos tht in th ginning, th rgistr ws rst to som unknown vlu W ssum th swithing tivity rom th unknown vlu to is som onstnt vlu Thn th ollowing is th hin o th t trnsitions! Y! Z! W I th input vril vlus r known, thn th xt swithing tivity is lult s onstnt H(; Y )H(Y; Z) whr H(i; j) is th Hmming istn twn two numrs i n j I, howvr, th iruit hs vn on input rnom vril, th whol systm hs to sri in proilisti wy s sri nxt Assum tht th n primry input rnom vrils r ; 2;:::; n n st A = ( ; 2;:::; n)g is th st ontining ll possil omintions o input tupls Lt st = (x; y) j x = x( ; 2;:::; n); y = y( ; 2;:::; n); 8( ; 2;:::; n) 2Ag Th swithing tivity twn th two onsutiv t vlus n Y is thn givn y: swithing(; Y ) = (x;y)2 i j k xy(x; y) H(x; y) () whr th summtion is ovr ll possil pttrns o (x; y) 2, n th untion H(x; y) is th Hmming istn twn two numrs x n y whih r rprsnt in rtin numr systm in inry orm Eqution ( ) rquirs tht th isrt typ joint p or x; y known Th mtho or lulting th joint p o two rnom vrils sri in stion 2 is minly suitl or th s whn th vrils in th systm r o ontinuous typ Whn howvr th prision us to rprsnt th isrt numrs is high nough or th vrin o th unrlying istriution is not too lrg, th ontinuous typ p g xy(x; y) n us s goo pproximtion or th isrt typ p P xy(x; y) tr ing multipli y th sling tor ( (x;y)2 gxy(x; y)) Th symoli omputtion mtho is howvr not vry prtil us it involvs th tsks o ning th symoli invrs solution o th systm o nonlinr qutions n symoli or numril intgrtion o omplit xprssions ovr th rgion n y omintion o inqulitis n/or qulitis Fortuntly, th sm swithing tivity or pir o isrt rnom vrils x n y n otin muh mor sily y th ollowing: swithing(;y ) = 2 n ( ; 2;:::; n) H(x( ; 2;:::; n);y( ; 2;:::; n)) (2) whr ( ; 2;:::; n) is th joint p o th input vrils ; 2;:::; n oth qution ( ) n qution ( 2) strt rom th ssumption tht th joint p ( ; 2;:::; n) is otin or known This is nssry onition in orr to prisly lult th ross-r swithing tivitis Furthrmor, qution ( 2) n us irtly on th input vtors r givn without otining th symoli xprssion or ( ; 2;:::; n) Hr w ssum tht th it with o rgistr is nit, so th totl numr o th pttrns tht n stor in rgistr is lso nit I w ssum ll o th numrs in our systm r intgrs (positiv or ngtiv), thn th totl numr o irnt (x; y) pirs involv in qution ( ) is t most 2 2it with In gnrl, qution ( 2) involvs multiimnsionl nst summtions ovr intrvls o intgrl vlus Whn th joint p o primry input vrils is n-limit (g Gussin), w n nrrow own th intrvl o summtion in h imnsion n thry signintly sp up th omputtion Lt's not th st A = ( ; 2;:::; n)g, st = (x; y) j x = x( ; 2;:::; n); y = y( ; 2;:::; n); 8( ; 2;:::; n) 2Ag, = (y; z) j y = y( ; 2;:::; n); z = z( ; 2;:::; n); 8( ; 2;:::; n) 2Ag, n D = (z;w) j z = z( ; 2;:::; n); w = w( ; 2;:::; n); 8( ; 2;:::; n) 2Ag Th totl swithing tivity in th ov xmpl with rgistr R shr y our rs (t vlus) is ormult s ollows: onstnt onstnt (x;y)2 (y;z)2 (z;w)2d xy(x; y) H(x; y) yz(y; z) H(y; z) zw(z;w) H(z;w) = ( ; 2;:::; n)(h(x; y) 2 n H(y; z)h(z;w)) (3) Th totl swithing tivity or rgistr n lult tr th th st o vrils tht shr tht rgistr

4 r oun Not tht th squn o t trnsitions r known t tht tim 3 Powr Optimiztion 3 Mx-ost ow ormultion Dnition 3 A irt grph G = (V,A) is ll th omptiility grph or rgistr llotion prolm i th it is onstrut y th ollowing prour Eh r (t vlu) i in th t ow grph hs n intrvl (irth tim i; th tim i) ssoit with it Eh opn intrvl i orrspons to vrtx i in G = (V,A) Thr is irt r(u; v) 2 A i n only i intrvl u \ intrvl v = ; n th tim u < irth tim v All proos n oun in [2] Thorm 3 Givn t ow grph without loops n rnhs, th omptiility grph G = G(V,A) or rgistr llotion prolm is yli Dnition 32 [5]An unirt grph G =(V,E) is omprility grph i thr xists n orinttion (V,F) o G stisying F \ F = ;; F F = E; F 2 F whr F 2 = (; ) j (; ); (; ) 2 F or som vrtx g omprility grphs r lso known s trnsitivly orintl grphs n prtilly orl grphs Dnition 33 Th unorint omptiility grph G = (V; E) is otin y rmoving th g orinttions o G =(V; A) Thorm 32 Givn t ow grph without loops n rnhs, th unorint omptiility grph G = (V,E) or rgistr llotion prolm is omprility grph To minimiz th totl powr onsumption on th rgistrs, ntwork N G = (s; t; V n;e n;;k) is onstrut rom th omptiility grph G = G(V; A) This is similr onstrution to th on us in [7] to solv th wight moul llotion prolm whih simultnously minimizs th numr o mouls n th mount o intronntion n to onnt ll mouls onptully, N G = (s; t; V n;e n;;k) is onstrut rom G = G(V;A) with two xtr vrtis, th sour vrtx s n th sink vrtx t Th itionl rs r th rs rom s to vry vrtx in V o G(V;A), n rom vry vrtx in V o G(V; A) tot W us th Mx-ost Flow lgorithm on N G to n mximum ost st o liqus tht ovr th G = G(V; A) Th ntwork on whih th ow is onut hs th ost untion n th pitis K - n on h r in E n Assuming tht h rgistr hs n unknown vlu t tim t,w us onstnt sw to rprsnt th swithing(unknown; v) or h vrtx v Mor ormlly, th ntwork N G = (s; t; V n;e n;;k) is n s th ollowing: V n = V [s; tg E n = A [(s; v); (v; t) j v 2 V g w(s; v) = L sw M (4) w(u; v) = L = L (u;v)2 2 uv(u; v) H(u; v) M ( ; 2;:::; n) H(u( ; 2;:::; n);v( ; 2;:::; n)) (5) w(v; t) = L; 8 v 2 V; w(t; s) = L: (6) whr A = ( ; 2;:::; n)g, = (u; v) j u = u( ; 2;:::; n); v = v( ; 2;:::; n); 8( ; 2;:::; n) 2Ag, L = mx swithing(u; v)g M ovr ll possil u,v 2 V [sg, n M is lrg onstnt us to sl up th smllst swithing tivity vlu to n intgr For h r 2 E n, ost untion : E n! N is n, whih ssigns non-ngtiv intgr ost to h r Th ost untion or ntwork N G is : (u; v) = w(u; v) or ll (u; v) 2 E n Th ost untion is n to init th powr svings on th r For h r 2 E n, pity untion K: E n! N, is n tht ssigns to h r non-ngtiv numr Th pity o ll th rs is on, xpt or th rturn r rom t to s whih hs pity k, whr k is usr-spi ow vlu n K(u; v) = ; 8(u; v) 2 E n n(t; s)g K(t; s) = k For h r 2 E n,ow untion : E n! N is n whih ssigns to h r non-ngtiv numr Th ow () on h r 2 E n must oy th ollowing: () K() n th ow onhvrtx v 2 V n must stisy th ow onsrvtion rul Thorm 33 A ow : E n! N with j j =, in th ntwork N G orrspons to liqu in th unorint omptiility grph G Thorm 34 A ow : E n! N, with j j = k, inth ntwork N G orrspons to st o liqus ; 2;:::; k in th unorint omptiility grph G Th gnrt liqus my not vrtx isjoint us th k pths in th N G my not vrtx isjoint On wy to nsur tht th rsulting liqus r vrtx isjoint is to mploy no-splitting thniqu This thniqu uplits vry vrtx v 2 V in th grph G = G(V;A) into nothr no v Thr is n r rom v to v or h v 2 V I thr is n r (u; v) 2 A in th grph G = G(V; A), thr is n r (u ;v) in th nw ntwork NG Thr is lso n r rom th sour vrtx s to vry vrtx v 2 V n rom vry uplit vrtx v to th sink vrtx t Mor ormlly, th no splitting thniqu gnrts th ollowing ntwork NG = (s; t; Vn;E n; ;K ) whr: Vn = V n [ V thr is vrtx v = (v) 2 V or h vrtx v 2 V En = A [(s; v); ((v);t);v 2 V g[(t; s)g [(v; (v) j v 2 V g

5 Dt Folw Grph / * s Ntwork Atr Vrtx Splitting t omptiility Grph s t Ntwork or Vrtx Splitting Figur 2: From t ow grph to ntwork N G A = ((u);v) j(u; v) 2 Ag ((t; s)) = ((v; (v)) = L; 8 v 2 V ((u ;v)) = ((u; v)) or ll (u ;v)2a [(s; v) ; ((v);t)j v 2 V g K ((t; s)) = k; K ((u; v)) = or ll u 6= t; n v 6= s: Th trnsormtions rom th t ow grph to th nl ntwork N G r shown in Fig 2 Thorm 35 A ow : E n! N, with j j = k, in th ntwork NG orrspons to st o vrtx isjoint liqus ; 2;:::; k in th unorint omptiility grph G Dnition 34 [4]Lt N = (s,t,v,e,,k) ow ntwork with unrlying irt grph G=(V,E), wighting on th rs ij 2 R or vry r (i,j) 2 E, pity K() or vry r 2E, n ow vlu v 2 R Th min-ost ow prolm is to n sil s-t ow o vlu v tht hs minimum ost In th orm o n LP: min t A = v vry no vry r vry r whr A is th no-r inin mtrix n ( i=s i = i=t othrwis Dnition 35 Th mximum ost ow prolm is tht givn ntwork N=(s,t,V,E,,K) n x ow vlu v, n th ow tht mximizs th totl ost Th sist mtho to solv th mx-ost ow prolm is to ngt th ost o h r in th ntwork, n run th min-ost ow lgorithm on th nw ntwork [4] Th prvious ntwork onstrution NG nsurs tht th rsulting pths r vrtx isjoint liqus in G (or G ) Whn th mx-ost ow lgorithm is ppli on this ntwork,w otin liqus tht mximiz th totl ost Th ow vlu on h pth is on, this implis tht th totl ost on h iniviul pth is th sum ovr ll iniviul rs on tht pth oring to thir topologil orr in th grph G = G(V; A), whr th ost on h r is linr untion o th \Sv Powr" For xmpl, i (s; ), (; ), (; ), (; t) is pth rom sour s to sink t Th totl ost on this pth is ost(s; ) ost(; ) ost(; ) ost(; t) Also, rom th ov inormtion, w n onlu tht th st o vrils ; ; g will shr th sm rgistr oring to th orr!! : Thorm 36 Th mx-ost ow lgorithm on th ntwork NG givs th minimum totl powr onsumption on th rgistrs in th iruit rprsnt y th omptiility grph G P Proo: Th totl ost is () (), whih is 2En linr untion o th \Totl Sv Powr" Th rson is tht 2 En L () () = 2En 2En () [L M swithing()] = () M 2En () swithing() In our spilly onstrut ntwork, () invry r xpt (t; s) P hs vlu ithr zro or on Th rst trm in th ov, (), is onstnt (=2jV jk 2En or G = G(V;A)) mong ll possil liqu ovrings tht ovr ll o th vrtis in th originl grph G Whn w mximiz th totl ost or givn ow vlu in NG,w r in minimizing th totl powr onsumption givn tht th numr o rgistrs is qul to this ow vlu Not tht, th mx-ost ow onnglwys ns th liqu ovring tht ovrs ll o th vrtis in th originl grph G whnvr th ow vlu j j is lrgr thn or qul to k min k min n trmin y th lt g lgorithm [] or simply y ning th mximum numr o rs tht ross ny -stp ounry In most ss, th k min oun y th lt g lgorithm is qul to th k min or mx-ost ow Howvr, in som pthologil ss, th two vlus r not th sm In tht s, post-prossing stp is n [2] 2 Th tim omplxity or th mx-ost ow lgorithm is O(km 2 ), oring to [4], whr m =2jV j2 or th grph G = G(V; A) n k is th ow vlus onitionl rnhs n sily hnl in our systm y rlxing th onitionl t ow grph into svrl

6 unonitionl t ow grphs n prorming th ov mtho on th iniviul rlx t ow grphs Du to th limit sp, w o not prsnt th tils hr A til xposition is provi in [2] 4 An xmpl Th ollowing xmpl is s on shul t ow grph s th on shown in Fig 3 This simpl t ow grph hs v primry input vrils,,, n For th sk o prsnttion, whoos th 5-vrit joint Gussin istriution s th joint p o,,, n Not howvr tht our mtho works or ritrry joint p's Th 5-vrit Gussin is goo hoi s this p is ommonly nountr in mny pplition omins, lik DSP Lt = A ; whr A = Th 5-vrit Gussin istriution is givn y: : : : : : () = Gussin5(; ; ; ; ) = p (2)5 t() xp 2 t g A whr mtrix is th ovrin mtrix or th 5-vrit Gussin joint p W ssum tht in this s it is givn y: 69:39 25:37 38:29 2:669 6:576 25:37 73:235 :992 4:952 63:63 38:29 :992 9:6 2:767 2:559 2:669 4:952 2:767 66:73 :366 6:575 63:63 2:559 :366 74:76 A Th numril vlus on th right hn si r provi s n xmpl From Fig 3, w know tht thr r six intrmit vrils in th t ow grph, tht is, rnom vrils, g, h, i, j, n k = g = h = ( ) ( ) i = ( )() j = 2 k = 2 ()() Th vrils' li tims r: (; 2); (; 2); (; 2); (; 2); (; 4); (2; 3); g(2; 4); h(2; 4); i(4; 5); j(4; 5); k(5; 6)g From th ov li tims, th prour us to onstrut th orint omptiility grph or th rgistr llotion prolm gnrts G = G(V; A) In this xmpl, w ssum 6-it rgistrs n th two's omplmnt rprsnttion or th vlus All numrs r in th rng [-32768, 32767] g * g h i ^2 j k Figur 3: Th shul t ow grph W us qution ( 2) in Stion 22 to lult th ross-r swithing tivitis or vry pir o rs in G(V;A): Th swithing tivity o or ny vril x rom tim = t whih is ssum to hv som unknown vlu to th tim tht th vril gts its rst vlu ws tkn to onstnt qul to (=5)[swithing(;)swithing(;) swithing(;)swithing(;)swithing(;)]: Atr lulting th swithing tivitis, w onstrut th mx-ost ow ntwork Th wight onh r is lult y qution ( 4)-( 6) in Stion 3 Hr w hoos M =, n so L = 837 Th ollowing wights r otin: w(x; x )=L; 8x 2 V ; w(s; x) = 527; 8x 2 V n othr w's r givn y th ollowing tl: g h i j k t g h i j Applying th mx-ost ow on th ntwork N G with th vrtx splitting thniqu, th ollowing rsults or numr o rgistrs, liqus n tul totl swithing tivity r otin: No o rg; liqus tot SA 7; ; g; ; g; ig; ; jg; g; g; hg; kgg ;; g; ; g; i; kg; ; jg; g; g; hgg ;; g; ; g; i; kg; ; hg; ; jg; gg 7882

7 Not tht our mtho ns th minimum powr rgistr ssignmnt or th givn numr o rgistrs To monstrt tht th swithing tivity lultion s on th joint p is nssry to otin low powr rgistr ssignmnt w prorm n xprimnt whr vry r wight in th omptiility grph ws st to som onstnt (= ) n thn rn th mx-ost ow or irnt ow vlus For ow vlu 5, w otin: Numr o rgistrs is qul to 5, liqus = ; h; kg; ; ; ig; ; g; jg; g; gg, tul totl swithing tivity = , whih is 355% wors thn th optimum solution Nxt, w gnrt rgistr ssignmnt solution using Rl [] whih ns th minimum numr o rgistrs n (in this s) n otin th ollowing rsult: Numr o rgistrs is qul to 5, liqus = ; ; i; kg; ; g; jg; ; hg; g; gg, tul totl swithing tivity = 7847, whih is 7% wors thn th optimum solution In, mong ll vli rgistr ssignmnt o givn siz, our propos lgorithm ns th on tht minimizs th powr onsumption Th prntg powr rution inrss or lrgr t ow grphs For xmpl, w otin 225% improvmnt in powr (ompr to th minimum rgistr ount rgistr ssignmnt prour) on 7-input t ow grph using similr ssumptions out th joint p n th t typs Spilly, For th Min-Powr Rgistr Assignmnt, w otin: Numr o rgistrs is qul to 9, liqus = ; ig; ; hg; ; j; kg; l; mg;ng; g; g;g; ggg n tul totl swithing tivity = 686; Numr o rgistrs is qul to 8, liqus = ; l; mg; ; hg;; j; kg; ;ig; g;g;gg;ngg n tul totl swithing tivity = 7272; Numr o rgistrs is qul to 7, liqus = ; l; m; ng;; hg; ; j; kg, ;ig;g;g; ggg n tul totl swithing tivity = 7763 For th Min-ount Rgistr Assignmnt, w otin: Numr o rgistrs is qul to 7, liqus = ; j; mg; ; h; k; ng; ; i; lg; g; g; g; ggg n tul totl swithing tivity = 7 5 onlusion This ppr prsnt novl wy to lult th swithing tivity xtrnl to st o rgistrs s on th ssumption tht th joint p o th primry input rnom vrils is known or n lult For shul t ow grph without yls, th omptiility grph or rgistr llotion n ssignmnt prolm ws provn to trnsitivly orintl grph A spil ntwork ws thn onstrut rom th ov omptiility grph n th mx-ost ow lgorithm ( vrition o min-ost ow lgorithm) ws prorm to otin th minimum powr onsumption rgistr ssignmnt Du to proprtis o trnsitivly orintl grph, th tim omplxity is polynomil Our utur work will ous on th rgistr ssignmnt or piplin sign n t ow grph with outr loops Aknowlgmnt Th rst uthor knowlgs hlpul isussions with hin-hi Hsu, Tung-Shn Lin n Kuo-hun L rom prtmnt o EE-systms t th Univrsity o Southrn liorni Rrns [] M lkrishnn n P Mrwl \Intgrt Shuling n ining: A Synthsis Approh or Dsign Sp Explortion" In pro o th 26th AM/IEEE DA, pp 68-74, 989 [2] J-M hng, M Prm,\Powr Eint Rgistr Assignmnt", ENG Thnil Rport 95-3, Univrsity o Southrn liorni, Mr 995 [3] G D Mihli,\Synthsis n Optimiztion o Digitl iruits"', MGrw Hill 994 [4] J Emons n RM Krp,\Thortil Improvmnts in Algorithmi Einy or Ntwork Flow Prolms", Journl o th AM, vol 9, no2, pp , April 972 [5] MGolumi, \Algorithmi Grph Thory n Prt Grphs", Ami Prss 98 [6] G Goossns, D Lnnr, J Vnhoo, J Ry, J Vn Mrrgn, n H D Mn \Optimiztion-s Synthsis o Multiprossor hips or Digitl Signl Prossing with thrl- II" In G Suir n PM MLlln, itor, Logi n Arhittur Synthsis, 988 [7] Y Hithok n DE Thoms, \A Mtho o Automti Dt Pth Synthsis", in Pro o th 2th Dsign Automti on Nw York, NY:AM/IEEE, pp , Jun 983 [8] R Hogg, A rig, \Introution to Mthmtil Sttistis" ourth Eition pp47-52 [9] D Ku, G D Mihli, \Rltiv Shuling Unr Timing onstrints", in DA, Proings o Dsign Automtion onrn, pp 59-64, Jun 99 [] K Kuukkr n A Prkr \MAAL, A Sotwr Pkg or Moul n us Allotion" Intrntionl Journl o omputr Ai VLSI Dsign, pp , 99 [] F Kurhi, A Prkr, \REAL: A Progrm or REgistr Allotion", in Proing o 24th Dsign Automtion onrn, Jun 987 [2] P Mihl, U Luthr, P Duzy, \Th Synthsis Approh to Digitl Systm Dsign", Kluwr Ami Pulishrs [3] M Pngrl n DD Gski, \Dsign Tools or Intllignt Silion ompiltion", IEEE trns on AD, 6(6), Nov 987 [4] Ppimitriou, K Stiglitz, \omintoril Optimiztion, Algorithms n omplxity" Prnti-Hll, in, 982 [5] A Ppoulis, \Proility, Rnom Vrils, n Stohsti Prosss" Thir Eition, stion 63, MGrw-Hill, 99 [6] PG Pulin n JP Knight,\Shuling n ining Algorithms or High-Lvl Synthsis", in Pro o th 26th AM/IEEE Dsign Automtion onrn (DA), pg -6, 989 [7] L Stok,\An Ext Polynomil Tim Algorithm or Moul Allotion", in Fith Intrntionl Workshop on High-Lvl Synthsis, uhlrhoh, pp69-76, Mrh [8] -J Tsng n DP Siwiork, \Automt Synthsis o Dt Pths in Digitl Systms", IEEE trns on AD, 5(3), July 986